Random Fixed Point Theorems for Relaxed Asymptotic Contractions in Random Normed Modules

TL;DR

This paper introduces a new class of random fixed point theorems for relaxed asymptotic contractions in random normed modules, generalizing existing results and unifying various fixed point theorems.

Contribution

It defines random relaxed asymptotic contractions using dual quasi-metrics and proves the existence and uniqueness of random fixed points under broad conditions.

Findings

Established the existence and uniqueness of random fixed points for the new class of contractions.

Generalized and unified several deterministic and random fixed point theorems.

Proved convergence of iterates in the psilon,-topology.

Abstract

We introduce the notion of a random relaxed asymptotic contraction in the setting of random normed modules. The contraction condition employs two quasi-metrics that are built directly from the random operator: a lower quasi-metric which adaptively switches between a four-point minimum and the ordinary one-step distance, and an upper quasi-metric which takes the maximum of four fundamental distances. The bounds are allowed to depend on the iteration index and are required to converge locally uniformly almost surely to a Boyd--Wong function. Using the fibre decomposition method based on \(\sigma\)-stability and the local property, we show that any such mapping defined on an essentially bounded, \(\sigma\)-stable and \(L^0\)-closed set admits a unique random fixed point, and all iterates converge in the \((\epsilon,\lambda)\)-topology. Our result strictly generalizes the random analogue of Kirk's asymptotic contraction theorem and unifies several deterministic and random fixed point theorems under a single flexible framework.